How do you integrate #int e^xe^x# using substitution?
Method 1 - Immediate Substitution
Method 2 - Simplification, then Substitution
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To integrate ( \int e^x e^{x} ) using substitution, let ( u = e^x ). Then, ( du/dx = e^x ) or ( du = e^x dx ). Substituting ( u ) and ( du ), the integral becomes ( \int u , du ).
Now integrate ( u ) with respect to ( u ), which gives ( \frac{u^2}{2} ). Then, resubstitute ( e^x ) for ( u ), yielding ( \frac{(e^x)^2}{2} + C ), where ( C ) is the constant of integration.
So, ( \int e^x e^{x} , dx = \frac{(e^x)^2}{2} + C ).
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To integrate (\int e^x e^{2x} , dx) using substitution, let (u = e^x). Then, (du = e^x , dx).
Substitute these into the integral: [\int e^x e^{2x} , dx = \int u e^{2x} , du]
Now, the integral is in terms of (u), so we can integrate it with respect to (u): [= \int u e^{2x} , du] [= \int u e^{2x} , du] [= \int u , du] [= \frac{1}{2} u^2 + C]
Finally, substitute back for (u) to get the final result: [= \frac{1}{2} (e^x)^2 + C] [= \frac{1}{2} e^{2x} + C]
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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